Quadrilateral Sides Proportional Section

[MilanVlasak]

Sorry guys, I thought it would be easy, but it is not.

I have a random quadrilateral A,B,C,D, drawn and a random point M inside of it.
Now, I want to draw a line passing through the point M and crossing the sides AB and DC at points P and Q.
So far so good, but I want more, in addition the line is required to be constrained by this ratio of distances:

AB:DC=AP:DQ

 

[MilanVlasak]

I got some progress here...

Definition:
If any line bisects the oppozite sides of a Qudrilateral A, B, C, D in points P and Q in such a manner the ratio AB:DC=AP:DQ holds than this line is called a Sweeping Line.

Theorem by Milan Anthony Vlasak:

"There is only one Parabola which is tangent to all Sweeping Lines of a Quadrilateral."

Any objection to that ?
You guys are mathematitions, I am a surveyor.

Milan

 

[MilanVlasak]

I can now EDIT the Definition:

Edited Definition:
If any line INTERSECTS the oppozite sides of a Qudrilateral A, B, C, D in points P and Q in such a manner the ratio AB:DC=AP:DQ holds than this line is called a Sweeping Line.

 

[0rthodontist]

If you extend AD and BC until they meet at point R, then RM is a sweeping line.

I don't think any parabola could be tangent to all sweeping lines of a quadrilateral.

 

[MilanVlasak]

Orthodontist,

If a line connects the midpoint of AB and the midpoint of CD than extention of this line will not generally pass the intersection point R of AD and BC.

Such midpoint-midpoint line is still a sweeping line according to my definition of a sweeping line: AB:DC=AP:DQ

(Nothing to do with collineation, cross ratio and Papus theorem)

Milan